2026年7月27日-8月7日，南开大学数学科学学院将举办南开大学数理逻辑暑期学校。我们非常荣幸的请到了巴黎西岱大学的 Boban Velickovic 和麦吉尔大学的 Marcin Sabok。欢迎大家参加。
Summer School
1.授课教师
Week 1: Boban Velickovic
Boban Velickovic is a Professor (Exceptional Class) of Mathematics at the University of Paris Cité. He obtained his PhD in 1986 under Prof. K. Kunen from the University of Wisconsin, Madison. Before coming to Paris he has had position at various universities in North America: Caltech, UC Berkeley, Carnegie Mellon University, etc. Prof Velickovic has made fundamental contributions to combinatorial set theory, the study of forcing axioms, and applications of set theory to various other areas of mathematics (measure theory, general topology, homological algebra, operator algebras, etc). He is one of the founding members of the European Set Theory Society (ESTS) and was its Vice President (2016-2018) and President (2019-2021). Prof. Velickovic has supervised a number of successful PhD thesis, two of his PhD students obtained the Sacks prize for the best thesis in mathematical logics given by the Association of Symbolic Logic.
Week 2: Marcin Sabok
Marcin Sabok received his PhD from University of Wrocław at 2009 and is now an Associate Professor at McGill University. His research includes classical set theory and functional analysis. His works have been published in several first rate mathematical journals including Inventiones Mathematicae.
Abstract: Game theory is the study of mathematical models of strategic interactions. First discussions of games can be traced back to Cardano in the 16th century. The foundations of game theory as an independent field were established by von Neumann in his paper On the Theory of Games of ******** in 1928. Game theory has important applications in many fields of social science, and is used extensively in economics. In this mini course we will concentrate on the interactions of game theory and mathematical logic and to a smaller extent computer science. We will discuss three fundamental games of logic: the evaluation games, the model existence game and the Ehrenfeucht-Fraissé game and show that they are closely related. We will use the game theoretic approach to prove some fundamental theorems of mathematical logic: the completeness theorem, Lowenheim-Skolem theorem, various interpolations theorems, etc. The approach can be adapted to some infinitary logics, such as . Program: Day 1: Preliminaries: ordinals, cardinals, the Axiom of Choice, Games of perfect information, mathematical concept of a game, infinite two player games, Gale-Stewart theorem on the determinacy of closed games. Day 2: Graphs, first order theory of graphs, Ehrenfeucht-Fraissé games on graphs, back-and-forth sets, general Ehrenfeucht-Fraissé and elementary equivalence.Day 3: First order logic, characterizing elementary equivalence, the Lowenheim-Skolem theorem, semantic games, the model existence game, interpolation theorems.Day 4: Infinitary logic: syntax and semantics of infinitary logic, dynamic Ehrenfeucht-Fraissé games.Day 5: Model theory for infinitary logics, Lowenheim–Skolem Theorem for , Model theory for , interpolation theorems for . More general infinitary logics. Reference: Models and games, J. Jouko Väänänen, Cambridge University Press, Cambridge Studies in Advanced Mathematics vol. 132 (2011)
Amenability and hyperfiniteness
Marcin Sabok
Abstract: During the series of lectures we will explore the notions of amenability and hyperfiniteness in the context of groups, graphs and equivalence relations. Among other things, we will see how algebraic properties of amenable groups are connected with the structure of their actions on probability spaces. We will discuss applications of that structure in probability and graph theory. Day 1. Groups, group actions and equivalence relations. Probability measure preserving (pmp) actions, ergodicityDay 2. AmenabilityAmenable groups, the Hulanicki-Reiter condition, free and pmp actions of non-amenable groups, ends of groupsDay 3. HyperfinitenessHyperfinite graphs and equivalence relations, Bernoulli graphs and factors of iid, Benjamini-Schramm limitsDay 4. CostGraphings, cost of groups, Hjorth's lemma on cost attainedDay 5. ApplicationsMatchings in hyperfinite graphs, applications to Poisson processes, circle squaring.
Referenes:
Kechris, Alexander S. Global aspects of ergodic group actions. No. 160. American Mathematical Soc., 2010.
Gao, Su. Invariant descriptive set theory. Chapman and Hall/CRC, 2008.
